A PD-Like Self-tuning Fuzzy Controller Without Steady-state Error

ELSEVIER Fuzzy Sets and Systems 87 (1997) 141-154

FiUZ2Y sets and systems

A PD-like self-tuning fuzzy controller without steady-state error

Chun-Tang Chao*, Ching-Cheng Teng Institute of Control Engineering, National Chiao-Tung University, Hsinchu, Taiwan

Received January 1995; revised January 1996

Abstract

This paper presents a PD-like self-tuning fuzzy controller (STFC) based on the tuning of scaling factors. A two-stage tuning method including a direct tuning stage and indirect tuning stage is proposed. For most linear controlled plants, application of the direct tuning stage alone is enough to yield satisfactory system responses. The proposed STFC can automatically detect the operating ranges of input variables and then adjust the scaling factors. In the indirect tuning stage the back-propagation algorithm can be directly applied to fine-tune the scaling factors when a more complicated plant is controlled. Furthermore, a simple but efficient method for fully compensating for steady-state error, which is the main problem in a PD-like fuzzy logic control system, is proposed so that tuning for fuzzy logic rule bases is unnecessary. Simulation results with linear time-invariant systems and a nonlinear unstable system as the controlled plants show that the proposed technique yields zero steady-state error responses very quickly without overshoot or oscillatory behavior. @ 1997 Elsevier Science B.V.

Keywords: Control theory; PD controller; Fuzzy controller; Scaling factor; Steady-state error

1. Introduction

The concept of a fuzzy set can be directly attributed to the seminal work o f Zadeh in 1965 [24]. Inspired by Zadeh, Mamdani and his associates [13] used a rule-based system with fuzzy parameters to construct a controller that emulated the performance of a hu- man operator. Sugeno, who appreciated the power of this new fuzzy rule-based paradigm for building con- trollers, began developing applications for this new methodology [ 19]. The main advantage of the fuzzy logic controller (FLC) is that it can be applied to plants that are difficult to model mathematically, and the controller can be designed to apply heuristic rules that reflect the experience of human experts.

* Corresponding author.

The first step in the design of an FLC is to deter- mine the input and output variables of the FLC [22]. In this step, the designers determine whether they will be using a PI-like, PD-like, or PID-like FLC. Since it is unrealistic to expect that an operator or expert can formulate reasonable control rules, considering third and higher dimensions, most common FLCs are PI-like or PD-like controllers [22]. Though the FLC exhibits superior applicabil i ty to the traditional PID controller [5, 20] and is highly robust [3], PI-like and PD-like FLCs possess mainly the same characteristics as traditional PI and PD controllers, respectively. That is, the PI-like FLC adds damping to a system and re- duces steady-state error, but yields penalized rise time and settling time. The PD-like FLC adds damping and reliably predicts large overshoots, but does not im- prove the steady-state response.

0165-0114/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH S01 65-011 4(96)00022-X

142 C.-T C77ao, (2-(2 Tenq/Fuzzy Sets and Systems 87 (1997) 141 154

Much research on the PD-like FLC has either con- sidered only simulation examples with no steady-state error problem [4] or reduced the steady-state error by fine tuning the rule bases, performing parameter optimization, and increasing the number of rules [8]. On the other hand, though the PI-like FLC can solve the steady-state error problem, techniques such as scaling-factor adjustment, rule modification [ 12], and membership-function shifting [25] are required in order to reduce the rise time and improve oscillatory behavior in the step response.

Once the membership functions and the rule-base of the FLC are set up, the next problem related to its im- plementation is the issue of tuning. The scaling factors (SF) are the main parameters used for tuning the FLC. If we demonstrate the virtual PI (or PD) approxima- tion of the PI-like (or PD-like) FLC, we can find that variations in the SFs result in modification of the poles of the overall transfer function and its zero [7]. This is the reason that changes in the SFs have a dramatic influence on the overall dynamics of the closed loop system. There is still no standard method for tuning the SFs. Thus an FLC that incorporates a systematic method for adjusting the SFs is urgently needed. Tun- ing rules for tuning the FLC by manipulating the SFs have been proposed in [6, 12, 15]. In these methods, however, the SFs can be tuned simultaneously only af- ter information such as rise-time, overshoot, degree of oscillation, settling time, and steady-state error is ob- tained. In [4], many tuning tables, in numerical form, were constructed for real-time simultaneous tuning of SFs. However, care must be taken in determining sev- eral parameters in [4].

In recent years there have been considerable de- velopments in the tuning of parameters in fuzzy logic systems [9, 10, 21 ] using the gradient-descent-based back-propagation (BP) algorithm [16], similar to methods in neural networks [ 14]. However, to use the BP method in the fuzzy logic systems, most authors construct the so-called neural-network-based fuzzy logic systems [10, 11,21]. We can find that it is time- consuming to take hundreds of learning epochs to train so many parameters in the fuzzy neural networks.

In this paper we favor the PD-like FLC, because it yields quick response with less oscillation than the PI- like FLC. Moreover, the problem of compensating for steady-state error in the PD-like FLC can be solved by a simple method in the proposed system. Owing to the

advantages of the proposed controller, the tuning pro- cedure for SF tuning is so simple that a direct tuning- stage alone is sufficient for the control of most linear plants. In the direct tuning-stage, the operating ranges of the input variables of the FLC can be detected auto- matically. When the controlled plant is more compli- cated, on the other hand, the BP algorithm is applied to adaptively tune the SFs in the indirect tuning stage. We do not need to construct a fuzzy-neural-network structure that there are only three parameters to be di- rectly tuned in this way. Furthermore, since the indi- rect tuning stage is based on the results obtained in the above direct tuning stage, only few learning steps are required. Simulation results indicate that the proposed self-tuning fuzzy controller (STFC) is indeed efficient.

The network structure and the tuning methods of the proposed STFC will be described in Section 2. A sim- ple method for compensating for the steady-state error in a PD-like FLC is presented in Section 3. Section 4 describes the proposed two-stage tuning method. The results of simulations conducted to evaluate the pro- posed STFC are presented in Section 5. Section 6 con- cludes the paper.

2. The proposed STFC

In this section, several tables are constructed that will be used in the tuning procedures in the proposed STFC. The procedure for detecting the operating ranges of the input variables and the supervised learn- ing used for adaptively tuning the SFs will also be described.

2.1. Modified decision table

A block diagram of a basic feedback control sys- tem is shown in Fig. 1. The purpose of the feedback controller under consideration is to maintain the out- put y ( k ) close to the set point sp. This is the so-called regulation problem. The definitions for the error e(k) and error change Ae (k ) are

e ( n ) = s p - - y(n), (1)

Ae(n) = e ( n ) -- e (n - - 1 ) = - - (y(n) y(n -- 1)),

(2)

C.-72 Chao, C.-C. Teng/Fuzzy Sets and Systems 87 (1997) 141-154 143

set- r ~ Y point

Fig. 1. Block diagram of a basic feedback control system.

Table 1 The control rules

E •

AE* NL NM NS Z PS PM PL

NL NL NL NM NM NS NS Z NM NL NM NM NS NS Z PS NS NM NM NS NS Z PS PS

Z NM NS NS Z PS PS PM PS NS NS Z PS PS PM PM PM NS Z PS PS PM PM PL PL Z PS PS PM PM PL PL

l NL N M NS ,_71:: PS PM PL

o

- 0 l

Fig. 2. Membership functions applied for the control rules.

y (t)

Fig. 3. Responses of process.

where indices n and n - 1 indicate the present state and the previous state of the system.

The control rules with two inputs and a single out- put fuzzy variables E*, AE*, and U*, representing e*(k), Ae*(k), and u*(k) of the controller, are shown in Table 1. We consider an FLC with the input and output SFs. The control rules in Table 1 are based on the characteristics of the step response [ 18]. The term sets of E*, AE*, and U* include the linguistic la- bels "positive large" (PL), "positive medium" (PM), "positive small" (PS), "zero" (Z), "negative small" (NS), "negative medium" (NM), and "negative large" (NL). The membership functions of the respective ref- erence fuzzy sets are plotted in Fig. 2. We combine the well-known min-max inference method cooperating with the popular center-of-area (COA) defuzzification procedure to produce the look-up table in Table 2, which will be referred to below as the decision table (DT).

The DT will be modified slightly so as to satisfy the two criteria shown below, where fixed SFs are considered.

1. For two points in the system response with the same e*, the point with the greater Ae* should have the greater u* in control action.

2. For two points in the system response with the same Ae*, the point with the greater e* should have the greater u* in control action.

We need to note that the word "greater" means "greater in signed value", e.g., 5 is greater than 3 and - 3 is greater than - 5 . It is reasonable that a PD- like FLC should obey these two criteria. The reason- ing behind these criteria is as follows. When plants are feedback-controlled, we can imagine obtaining the two responses shown in Fig. 3. Point A has the same error value as point B, but point A has a greater Ae than point B. A reasonable control output for point A should be greater than that for point B. On the other hand, suppose points C and D have the same error change, while the e for point C is significantly greater than that for point D. Then the control output for point C should be greater than that for point D. In the above cases, we have e > 0 and Ae < 0. The modified de- cision table (MDT) constructed by applying the above criteria is shown in Table 3, where the elements in bold-face have been modified from the original DT.

Instead of quantizing the scaling results as the in- dices to map an element in the MDT, the mapped el- ement is obtained by performing interpolation. From the MDT, the approximate equations for calculating

144

Table 2 The decision table (DT)

C.-T. Chao, C.-C. Tenfl/Fuzzy Sets and Systems 87 (1997) 141 154

e •

Ae* 1.2 -- 1.0 0.8 0.6 --0.4 --0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.2 0.95 0.95 --0.92 --0.80 --0.75 --0.67 0.54 0.50 --0.41 --0.25 --0.17 0.12 0.00 -1.0 -0.95 -0.8 -0.92 -0.6 --0.80 -0.4 -0.75 --0.2 -0.67

0.0 -0.54 0.2 -0.50 0.4 -0.41 0.6 -0.25 0.8 --0.17 1.0 0.12

-0.85 0.84 -0.82 -0.68 -0.56 -0.56 0.47 -0.30 -0.15 -0.07 -0.00 -0.84 0.75 0.75 -0.68 -0.52 -0.47 0.44 0.24 -0.09 0.00 0.07 -0.82 0.75 0.56 -0.56 -0.52 -0.34 0.24 0.12 --0.00 0.09 0.15 -0.68 -0.68 -0.56 0.40 -0.40 -0.26 -0.09 -0.00 0.12 0.24 0.30 -0.56 -0.52 -0.52 -0.40 0.20 -0.09 0.00 0.09 0.24 0.44 0.47 -0.56 -0.47 0.34 0.26 -0.09 0.00 0.09 0.26 0.34 0.47 0.56 -0.47 -0.44 -0.24 0.09 0.00 0.09 0.20 0.40 0.52 0.52 0.56 -0.30 -0.24 -0.12 -0.00 0.09 0.26 0.40 0.40 0.56 0.68 0.68 -0.15 -0.09 -0.00 0.12 0.24 0.34 0.52 0.56 0.56 0.75 0.82 -0.07 0.00 0.09 0.24 0.44 0.47 0.52 0.68 0.75 0.75 0.84

0.00 0.07 0.15 0.30 0.47 0.56 0.56 0.68 0.82 0.84 0.85

0.12 0.17 0.25 0.41 0.50 0.54 0.67 0.75 0.80 0.92 0.95

1.2 --0.00 0.12 0.17 0.25 0.41 0.50 0.54 0.67 0.75 0.80 0.92 0.95 0.95

Table 3 The modified decision table (MDT)

e*

Ae* -- 1.2 -- 1.0 --0.8 --0.6 0.4 --0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

--1.2 --0.95 0.95 --0.92 --0.80 --0.75 0.67 --0.54 --0.50 --0.41 --0.25 0.17 --0.12 --0.00 - l . 0 --0.95 --0.8 --0.92 --0.6 --0.80 --0.4 --0.75

0.2 --0.67 0.0 0.54 0.2 --0.50 0.4 -- 0.41 0.6 --0.25 0.8 --0.17 1.0 --0.12

-0.85 -0.84 -0.82 -0.69 -0.56 -0 .53 -0.47 -0.30 -0.15 -0.07 -0.00 -0.84 -0 .76 0.75 -0.68 -0.52 -0.47 -0.44 -0.24 -0.09 0.00 0.07 --0.82 --0.75 0.57 --0.56 --0.50 --0.34 --0.24 --0.12 --0.00 0.09 0.15 --0.69 --0.68 0.56 --0.41 --0.40 --0.26 0.09 --0.00 0.12 0.24 0.30 --0.56 --0.52 --0.50 0.40 --0.20 --0.09 0.00 0.09 0.24 0.44 0.47 --0.53 --0.47 --0.34 0.26 --0.09 0.00 0.09 0.26 0.34 0.47 0.56

0.47 --0.44 --0.24 -0.09 0.00 0.09 0.20 0.40 0.50 0.52 0.57 --0.30 --0.24 --0.t2 --0.00 0.09 0.26 0.40 0.41 0.56 0.66 0.68 --0.15 --0.09 --0.00 0.12 0.24 0.34 0.50 0.56 0.57 0.75 0.82 --0.07 0.00 0.09 0.24 0.44 0.47 0.52 0.66 0.75 0.76 0,84 --0.00 0.07 0.15 0.30 0.47 0.56 0.57 0.68 0.82 0.84 0.85

0.12 0.17 0.25 0.41 0.50 0.54 0.67 0.75 0.80 0.92 0.95

1.2 -0.00 0.12 0.17 0.25 0.41 0.50 0.54 0.67 0.75 0.80 0.92 0.95 0.95

~u*/Oe* a n d ~.u*/OAe* s h o w n b e l o w can be ob ta ined ;

t he se e q u a t i o n s wi l l be u s e d in the ind i rec t t u n i n g

s tage:

Ou* M D T ( A e * , e * + I ( e * ) ) - M D T ( A e * , e * )

#e* I(e* ) '

fo r I (e*) ~ 0 ( 3 )

Ou* M D T ( A e * + I ( A e * ), e* ) -- M D T ( A e * , e* ) OAe* I ( A e * )

fo r I ( Ae* ) --, O,

w h e r e I (x) is a f u n c t i o n to take little i n c r e m e n t o n x.

In the p r o p o s e d s y s t e m , w e a s s i g n I(e* ) = I( Ae* ) = 0.01. W e a lso find that the i n f e r r ed Ou*/cqe * and

~u*/?~Ae* are pos i t i ve , as r e q u i r e d b y c r i t e r ion 2 and

c r i t e r ion 1, r e spec t ive ly . M o r e o v e r , Eqs . ( 3 ) and ( 4 )

ind ica te tha t a D T or M D T w i t h in te rva l [ - 1 . 2 , 1.2]

is c o n v e n i e n t fo r p r o g r a m m i n g .

2.2. Layered operation

T h e p r o p o s e d f o u r - l a y e r n e t w o r k s t ruc tu re o f the

( 4 ) S T F C is i l lus t ra ted in Fig. 4. In th is s u b s e c t i o n , w e

C.-T. Chao, C-C. Teng/Fuzzy Sets and Systems 87 (1997) 141 154 145

S l o p e = s I = s e

e-~~_, ~

Slope = s 2 = sa

Z/* e*

-1 -0.8

0 0.2

1

-1 - 0 . 8 . . . 0 0 . 2 . . . 1

(or S 3 )

U

Layer 1 Layer 2 Input layer Scaled layer

Fig. 4. The network structure

shall describe the signal propagation and the basic function of the nodes in each layer. We use netj and J) to denote the summed net input and activation function of node j , respectively, and the superscript denotes the layer number. Moreover, x~ and y~ denote the input and output vector of the j th node in layer k, respectively.

L a y e r 1 : input layer. For the jth, j = 1,2, node of layer one, the net input and the net output are

net) : # and y) : f j l ( n e t ) ) : net) , (5)

where x I = e andx~ = Ae.

L a y e r 2: scaled layer. In this layer, the operating ranges of the measured variable e ( k ) and Ae(k) are transformed to the normalized universe [ - 1, 1 ] by the scaling factors Se and Sd, respectively. Before the layer two operation, an S F - a d j u s t m e n t procedure is per- formed:

1/e i f ese > 1 se - 1 / e i f e s e < - 1 (6)

1/Ae i f Aesd > 1 S d = - - 1 / A e i f Aesd < --1 (7)

The SF-adjustment procedure is crucial for the direct tuning stage, as will be explained in Section 4. We can find that the input scaling factors are changed only when the emax or Aemax is obtained. Thus, it seems

Layer 3 Layer 4 Table look-up layer Output layer

of the proposed STFC.

that we can predict the operating ranges O R e and ORd of e and Ae, respectively, by

ORe = [--e . . . . emax] = [--1/Se, l/se],

ORd = [--Aemax,Aemax] = [--1/Sd, 1/Sd].

After the SF-adjustment procedure, the operation in this layer is

net } : s i * x } and y } : # ( n e t } ) : n e t } . (8)

We note that sj = Se, s2 = sd, yf = e*, and y2 = A c * .

L a y e r 3: table look-up layer. In this layer, we per- form MDT-mapping for the scaling results e* and Ae*. Thus we have

net 3 = M D T ( x ~ , x ~ ) and (9)

y3 = u* = f 3 ( n e t 3 ) = ne t 3.

L a y e r 4: output layer. The mapped element in the last layer is also scaled by an SF, say s, (or s3), and is added by a constant gain Cg to arrive at the desired control signal. The final output of the network is

net 4 = su * x 4 ~- Cg and (lO)

y4 = U = f 4 ( n e t 4 ) = ne t 4.

The constant gain Cg is crucial for zeroing steady-state error, as will be explained in the next section.

146 C-72 Chao, C.-C. Ten(j/Fuzzy Sets and Systems 87 (1997) 141 154

2.3. Supervised gradient descent learning

Since the proposed STFC is a four-layer feedfor- ward network, it is a straightforward task to apply the gradient-descent-based BP algorithm [16] to adap- tively adjust the SFs. The goal is to minimize a cost function, E, so that training pattern k is propor- tional to the square o f the difference between the set point sp and the plant output y(k) . Let E be defined by

E = l ( s p - y ( k ) ) 2. ( 1 1 )

If si, i = 1,2, 3, is the adjusted SF, then the learning rule is

aE s i ( k 4- 1) = s i ( k ) - ~ i - ~ - -- °~iAs i (k ) ( 1 2 )

(ps i

and

A s i ( k ) = s i ( k ) - s i ( k - 1), (13)

where qz is the learning rate and c~i, 0 < ~i < 1, is the momentum parameter. We will now to derive the learning law for each layer in the feedback direction.

Layer 4: The gradient o f error in (11 ) with respect to an arbitrary weighting vector W E R ~ is

aE ,, a e ( k ) , , , a y ( k ) bSY - et,~) ? ~ -- etK~ V Y

, , , a y ( k ) au(k) SO(k) = - e t t c ) ~ ( ~ ~ - - e ( k ) y , ( k ) ~ - - , ( 1 4 )

where O(k) is the output of the STFC and S = yu(k) = ay (k ) / au (k ) is the plant sensitivity. From (14), we can derive the propagating error term given by the output node

34 _ _ - a E _ e ( k ) y , ( k ) ~ = e(k)yu(k) . (15) ane t 4 one t

Then, we have

c~E _ a E ay 4 anet 4 _ fi4y3 = 34u*. (16) 6qs3 a y 4 ane t 4 as3

Hence, by (12), the scaling factor s3 is updated by

s3(k + 1) = s3(k) + ~1364(k)y3(k) + :~3As3(k). (17)

Layer 3: The error term 33 is derived as follows:

~ 5 3 - a E _ - a E a y 3 _ - a E (18) anet 3 ay3 ane t 3 (~y3

-- -- 6~E cy4 -- 34s3 . (19) ay 4 0y 3

Layer 2: First, the error term is computed:

-2 - -aE - a E ay 2

a j - a n e t 2 - ay 2 anet 2

- a E {3y3 "3 aY 3 - ay- ay? - 3

,5 &-- for j = 1, ae*

33 au* for j = 2. aAe*

We can then derive

_ a E c~y~ _ a2 ayJ 2 x !

I

aE

as, a7 " aN : 3,. ,

{ 2 2' jX) for - 1 < y 2 < 1, forv 2 = l o r v 2 = 1.

Thus, the update rules for s j, j = 1,2, are

s j (k + l ) = s j (k ) + ~ja2x) + ~jasj(k).

(20)

(21)

(22)

3. A simple method for zeroing steady-state error

PD control can reliably predict and correct large overshoots, but the derivative control will affect the steady-state error of a system only if the steady-state error varies with time. i f the steady-state error of a sys- tem is constant with respect to time, the time deriva- tive of the error will be zero, and derivative control will have no effect on the steady-state error. In this pa- per we consider the steady-state error for a step input in a control system with PD control. We try to fully compensate for the steady-state error by a proposed simple method.

Basically, the proposed STFC in Fig. 4 is composed of a fuzzy PD controller and an added constant gain. In the following, we will show how to fully compensate for the steady-state error by this structure. From the operations in the table look-up layer, we shall demon-

C-T. Chao. C.-C. Teng/Fuzzy Sets and Systems 87 (1997) 141-154 147

WIs)

r ,~ ,~ .p-~o , t , r-p . . . . ~°1 ~ o~t, I . . . . . . . I -y , t ,

Fig. 5. Control system with PD control.

We will make the linguistic trajectory converges exactly to the origin with the help of the constant gain Cg. Considering the equilibrium point at x = [xa(t) x2(t)] T = [e(t) d(t)] T = 0, where the sys- tem output response is maintained at set point sp and the output of the fuzzy PD controller is zero, we have

strate the effect of the normalization on the virtual PD approximation as

u*(k) : / ( p e * ( k ) + KDAe*(k ). (23)

Thus, the following equality will be true for the real values:

u( k ) = suu* ( k ) + Cg

= Kp(suSe)e(k ) 4- [~D(SuSd)Ae(k) + Cg

= / £ p e ( k ) + / £ D A e ( k ) 4- Cg. (24)

We can transform the above discrete expression to its continuous form

u(t) = / £ p e ( t ) + TI£Dd(t ) + Cg, (25)

where T is the sampling period. A control system with fuzzy PD control, added by

a constanr gain, is shown in Fig. 5. Let Gp(s) denote the transfer function of the linear controlled plant. I f we first ignore the constant gain Cg, the DC gain from set point sp to the plant output yp is

G(O) -- KpGp(O) 1 + /~pGp(O) ' (26)

where G(s) is the transfer function of the overall sys- tem and Gp(0), Gp(0) > 0, is the DC gain of the plant. From (26), we can see that steady-state error due to a step-function input will always be present if Gp(0) --+ oc is not satisfied. From the viewpoint of the FLC, this means a trajectory converging to the origin (e = Ae -- 0) in linguistic space [1], with fuzzy variables E and A E as its axes, is impossible unless Gp(0) --* cx) is satisfied. When Gp(0) does not approach infinity, the linguistic trajectory of the overall control response will eventually reach a stable point in the neighbor- hood of the origin with a certain degree of steady-state error.

cgGp(O) =sp. (27)

We find that the value of Cg has no relationship with the control constant ga ins /£p or/£D. Also, it seems that if we set

sp (28) C g - Gp(0) '

we can solve the steady-state error problem. To show this, we can derive the output transfer function

Gp(s)( TI(DS 4-/~p) Y(s) = Gp(s)(TI~DS 4-/~p) Jr- 1R(s)

Gp(s)

-f Gp(s)(TI£Ds 4-/(p) 4- 1 W(s), (29)

where we can find that the step signal w(t) does not influence the stability of the overall system. By substi- tuting sp/s and Cg/S, respectively, for R(s) and W(s) in (29), we have the steady-state output

lim yp(t)- t ~ O C

Gp(0)/(p sp

Gp(0)Kp + 1

Gp(O) sp

Gp(0)Kp 4- 1 Gp(0)

=sp, (30)

with steady-state error ess = 0.

4. Two-stage tuning method

In Sections 2 and 3, we obtained an important fact that variations in the SFs implies the varia- tions in the operating ranges ORe, ORd, and ORu. It means that if we want to fine-tune SFs, the remaining problem is how to automatically detect the operat- ing ranges and then adjust the SFs. For brevity, we neglect the magnitude-constraint on the controller

148 C.-Z Chao, C.-C Teng/Fuzzy Sets and Systems 87 (1997) 141 154

output u(k). lnstead of tuning the SFs through trial- and-error, the proposed system employs a systematic approach:

1. In the direct tuning stage, the SFs are directly and iteratively adjusted to reduce the error between the plant output and the set point. The supervised learning described in Section 2.3 is not used in this stage.

2. In the indirect tuning stage, the BP algorithm is applied to fine-tune the SFs obtained in the above stage. This tuning stage can be omitted if the desired control result is achieved in the direct tuning stage.

In the direct tuning-stage, we first initialize Se and Sd with large values and then increase the value of su, starting from a low value, in each iteration. Generally speaking, when the value of s3 increases, the operating range of e(k) and A e ( k ) will expand since the varia- tion in system response is greater. The SF-adjustment procedure in (6) and (7) will detect this situation and reduce the value of se and sd. Thus, though only the layered operation is performed in this stage, the SFs will be automatically adjusted. Of course, se will be adjusted as

1 se = - - (31)

sp

in this tuning stage for most cases, except when the step response exhibits a small negative undershoot near the time index k = 0. Note that if the initial val- ues s~ and sd remain unchanged after the first iteration, this means that their values should be further enlarged. For most of the linear controlled plants studied in our simulations, this simple direct tuning stage is sufficient to provide the desired output. The user can record the performance indices such as the rise time, reach time, overshoot, and settling time after each iteration to de- termine whether the desired system response has been achieved.

Furthermore, in the first iteration of the direct tuning stage, we can assume that Gp(0) --+ oc and let Cg = 0. If we obtain the system output with nonzero steady- state error, we then do approximation on

G p ( 0 ) - Yss (32) Uss

by the steady-state input Uss and output Yss of the plant in the following tuning iterations. In this way, the proposed compensation method will be robust to varying sizes of DC disturbances of the controlled

plant. This approach is adopted in the simulation examples in Section 5.

If the desired control result is not achieved in the above stage, supervised learning can be applied in the indirect tuning stage, and a plant-identifier can be em- ployed to identify the plant sensitivity Yu to imple- ment the indirect adaptive FLC. We refer readers to reference [2], in which a FNN identifier is applied. It is not practical to simultaneously tune SFs Se, sa,

and su to obtain a desired response with freely chosen initial values, because the process may be trial-and- error and learning convergence problems may arise. It is feasible to perform supervised learning based on the SFs obtained in the first tuning stage, however, because in this case the tuning method is likely to converge to a desired solution. In the proposed sys- tem, micro-tuning of the results of the first stage is performed in the second stage if we set all learn- ing parameters to smaller values. On the other hand, varying the SFs by setting all learning parameters to larger values can result in considerable nonlinearity of the fuzzy controller, but convergence is hard to be guaranteed in this approach. Simulation results indi- cate that the two-stage tuning method is feasible and efficient.

5. Simulation examples

In the following simulation examples, suppose each learning cycle takes te seconds with a step size of ts

seconds.

Example 1 (An under-damped system). The plant to be controlled is described by the Laplace transfer function [8]

20 Gp(s ) - s 2 + 8 s + 2 0

Let te = 10, ts = 0.02, and sp = 4, we initialize se = Sd -- 1000 and su = 20. The value of su is in- creased by one in each iteration. Supervised learning is not applied here, so no plant-identifier is required. When steady-state error compensation method is applied, Table 4 shows values of the main parameters in the first five iterations. We find that the SFs are automatically tuned by the proposed SF-adjustment

C.-T Chao, C.-C. Teng/Fuzzy Sets and Systems 87 (1997) 141 154 149

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

/

i 2 3 4 5 6 7 8 9 10

Time (sec)

Fig. 6. Step response of the process with su 60 in Example 1. The dashed line indicates the system response before steady-state error compensation (Se = 0.25 and sd = 2.77); the solid line indicates the response after steady-state error compensation (Se 0.25 and sa = 2.357).

Table 4 Values of the main parameters in the first five iterations

Iteration Se Sd Su Cg Predicted ORe Predicted ORa yss

1 0.25 7.35 20.0 0.00 [-4.0,4.0] [-0.136, 0.136] 2.934 2 0.25 5.57 21.0 4.00 [-4.0,4.0] [-0.180,0.180] 4.000 3 0.25 5.02 22.0 4.00 [-4.0,4.0] [-0.199,0.199] 4.000 4 0.25 4.78 23.0 4.00 [-4.0,4.0] [-0.209,0.209] 4.000 5 0.25 4.62 24.0 4.00 [-4.0,4.0] [-0.216,0.216] 4.000

procedure. To illustrate the eff iciency o f the direct tun-

ing stage, Fig. 6 shows the step responses before and

after steady-state error compensa t ion when su = 60.

W e find that the original steady-state error o f 0.53

quickly decreases to zero. W e also find that the pro-

posed direct tuning me thod can automat ica l ly adjust se and sa by detect ing the operat ing ranges o r e ( k ) and A e ( k ).

E x a m p l e 2 (A d a m p e d oscillation s y s t em) . The con-

t rol led sys tem is a second-order sys tem mode l ed as

fol lows:

y ( t ) ÷ 0.40))(t) + 0 .54y( t ) = 19.54u(t).

This is a var ia t ion o f a vehic le speed control system

with a potential disorder [ 12]. Let te = 6, ts = 0.02,

and sp = 60. Then we can control the plant as wel l as that in Example 1. I f the p roposed compensa t ion me thod is not applied, most designers wou ld increase

the value o f su to reduce the steady-state error. Fig. 7

presents an example which shows that the steady-state

150 C.-7C Chao, C.-C. Teng/Fuzzy Sets and Systems 87 (1997) 141 154

7O

t 5O

40

30

20

10!

t 1 2 3 4 5

Time (sec)

Fig. 7. Step response of the process with su = 400 in Example 2. The dashed line indicates the system response before steady-state error compensation (Se = 0.017 and sa = 0.294); the solid line indicates the response aider steady-state error compensation (se 0.017 and sd = 0.290).

error cannot be completely eliminated even when s , is increased to 400. When the proposed method is applied, however, the steady-state error is very quickly reduced from 0.56 to zero.

Example 3 ( A t i m e - d e l a y s y s t e m ) . The transfer func- tion of the controlled plant is

1 G p ( s ) = s ~ - ~ e x p ( - 0 . 5 s ) ,

which involves a t ime delay. This example corre- sponds to situations often encountered in industrial processes. Let te = 20, ts = 0.02, and sp = 1. As in the above examples, we initialize Se = sa = 1000 and s~ = 1 and then increase the value of s~ by one in each iteration. Fig. 8 shows the step response before and after steady-state error compensation when s~ is 19. The system response with zero steady-state error again confirms the applicabil i ty and efficiency of the proposed system.

Example 4 (A non l inear uns tab le s y s t e m ) . The plant is described by the differential equation [23]

~ z 1 2 ~ y - y + u .

In this case we let te = 7.5, ts = 0.015, and sp =

2.0. In this example, the suggested two-stage tuning method is introduced because of the complexi ty of the controlled plant. In the direct tuning stage we initial- ize Se = sa = 1000 and su = 1 and then increase the value of su by one in each iteration. The mean-square error of the response gradually decreases as su in- creases until su = 39, at which time we have se -- 0.5 and sd = 2.79. The step response is indicated in Fig. 9 by the dashed line. Using the above result, we begin to perform micro-tuning of SFs in the indirect tuning stage. Note that the plant sensitivity Yu is calculated by numerical approximation. We set r/i = cq = 0.001 for i -- 1 to 3 and perform supervised learning. After 50 learning epochs, we obtain the desired system re- sponse shown in Fig. 9 by the solid line. The respec- tive learning trajectories of Se, sa, and s , in the 50th learning epoch are shown in Fig. 10.

C - T . C h a o , C . -C . T e n g / F u z z y S e t s a n d S y s t e m s 8 7 ( 1 9 9 7 ) 141 1 5 4 151

i

0.9 . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . • . . . . . . . . . . . . . . .

0.8 ,,." . . . . . . . . . . . i . . . . . . . . . . . .

0.7 ,': . . . . .

0 .6 ," . . . . . . . . . . . . . . .

• 0.5 i . . . . . : . . . . . . . . . . . . .

0.4 : . . . . . . . . . . . . . -

o.3 i ~

o . 2

o., °

0 2 4 6 8 10 12 14 16 18 20

T i m e (sec)

Fig. 8. Step response o f the p rocess wi th su = 19 in E x a m p l e 3. The dashed line indicates the sys t em response be fo re s teady-s ta te

er ror c o m p e n s a t i o n (Se = t .0 and Sd = 34.229) ; the solid line indicates the r e sponse af ter s teady-s ta te error c o m p e n s a t i o n (Se = 1.0 and

sd = 29 .492) .

2.5 !

2 . . . . . o ; , ,', 3 ,', ; , ,, ,~ ,'~ ;, ", ~ ,,,, i", ,' 1,, ,~ 1' , ," , ",

I i . . . .

0.5 : . . . . . . . . . .

i

}

0 ' i i i 0 O. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

T i m e (sec)

Fig. 9. Final sy s t em response for E x a m p l e 4. The response af ter d i rect tun ing s tage (Se ~ 0 . 5 , S d 2.79, and Su 39) is indica ted by

the dashed line, that a f te r the indirect tun ing s tage ind ica ted b y the sol id line.

152 C-T. Chao, C.-C. Teng/Fuzzy Sets and Systems 87 (1997) 141-154

(a) 2.4

2 .

1.8 . . . .

1 . 6 . . . . . . . . . . . . . . .

~ 1.4

1 . 2

l . . . .

o . 8 -

o . 6

0.4 . . . . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (sec)

0.18

0.16

0.14

0.12 _ H

0.1

0.08

0.06 0

, , , , , _ , , ,

0.1 0.2 0.3 0.4 0.5 O.6 0.7 0.8 0.9

Time (sec)

(b) 0.2

Fig. 10, The learning trajectories of the scaling factors in Example 4. (a) Se, (b) Sd, and (c) s~

C2-T. Chao, C.-C. Teng/Fuzzy Sets and Systems 87 (1997) 141-154 153

(c) 46.1

4 6 : : . i ! : i i ~ . . . . . . . . . . . . . . . . . . . . . i i

45.'i

45.,

45.7

45.6

45.5 . . . .

45.4

45.3

45.2

45.1 , i , i , i i 0 0.1 0.2 0.3 04. 0.5 0.6 0.7 0.8 0.9

Time (sec)

Fig. 10. Continued.

6. Conclusion

In this paper, we have described a PD-like self- tuning fuzzy controller (STFC) based on the tuning of scaling factors. We have solved the steady-state error problem in a PD-like FLC system by a simple but efficient method. Furthermore, to avoid tuning the SFs through trial-and-error, we have proposed a systematic two-stage tuning method. In the direct tuning stage, the proposed STFC automatically de- tects the operating ranges of the input variables and then adjusts them. In the indirect tuning stage the design results of the direct tuning stage are used to adaptively fine-tune the SFs if the desired result was not obtained in the previous stage. Simulation results for linear time-invariant systems and a nonlinear un- stable system show the proposed technique produces zero steady-state error responses very quickly without overshoot or oscillatory behavior.

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